Let $X=\mathbb{A}_{\mathbb{C}}^1\backslash\{0\}$. The first algebraic de Rham cohomology of $X$ is 1-dimensional, and has a canonical generator $\frac{dz}{z}$ (or any other 1-form with residue 1). The first Betti cohomology is also 1-dimensional, and has a canonical generator taking a 1-cycle to its winding number around 0. Over $\mathbb{C}$ there is a canonical isomorphism $H^1_{dR}(X)\to H^1_B(X,\mathbb{C})$, but it takes the generator to $2\pi i$ times the generator.

If we wanted to say this in terms of isomorphisms, we could consider the three vector spaces $\mathbb{C},H^1_{dR}(X)$, and $H^1(X,\mathbb{C})$, and use the fact that an isomorphism from $\mathbb{C}$ to a 1-dimensional vector space is the same thing as the choice of a generator of that vector space.